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The slope field of () = +, showing three of the infinitely many solutions that can be produced by varying the arbitrary constant c.. In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral [Note 1] of a continuous function f is a differentiable function F whose derivative is equal to the original function f.
Finding an elementary antiderivative is very sensitive to details. For instance, the following algebraic function (posted to sci.math.symbolic by Henri Cohen in 1993 [3]) has an elementary antiderivative, as Wolfram Mathematica since version 13 shows (however, Mathematica does not use the Risch algorithm to compute this integral): [4] [5]
This visualization also explains why integration by parts may help find the integral of an inverse function f −1 (x) when the integral of the function f(x) is known. Indeed, the functions x(y) and y(x) are inverses, and the integral ∫ x dy may be calculated as above from knowing the integral ∫ y dx.
Here is a brief overview of what Xcas is able to do: [9] [10] Xcas has the ability of a scientific calculator that provides show input and writes pretty print; Xcas also works as a spreadsheet; [11]
When evaluating definite integrals by substitution, one may calculate the antiderivative fully first, then apply the boundary conditions. In that case, there is no need to transform the boundary terms. Alternatively, one may fully evaluate the indefinite integral first then apply the boundary conditions. This becomes especially handy when ...
In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an elementary function. [1] A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist. [ 2 ]
See antiderivative and nonelementary integral for more details. A procedure called the Risch algorithm exists that is capable of determining whether the integral of an elementary function (function built from a finite number of exponentials , logarithms , constants , and n th roots through composition and combinations using the four elementary ...
In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations.The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called collocation points), and to select that solution which satisfies the ...